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Robertson, JB, Uppuluri, VRR and Rajagopal, AK (1983)

First digit phenomenon and ergodic theory

Journal of Mathematical Analysis and Applications 95(2), pp. 375-378.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.

Abstract: For any k∈I+ = the set of positive integers, let S(k) denote the set of positive integers beginning with k, i.e., S(k) = { n: k = ⎣n/10r⎦for some r∈I+}, where ⎣x⎦ denotes the greatest integer less than or equal to x. Consider some finitely additive extension of the density function d(S) = limN→∞ |S∩[1, N]|/N, where S⊂I+ and |S| denotes the cardinality of S. If d(k) = d(S(k)) is defined for all positive integers k, and if the mapping T that maps S into (2S)∪(2S + 1) preserves the density d, then the result of Cohen is that d(p) = log10(1 + (1/p)) for p∈D. The crucial step here is the specific relation of the transformation T to the multiplication of the elements of S by 2. This motivates us to consider the following problem: Set b = 10 and let a∈R+ be a positive real number. Let θ(x) = ax, where x∈R+, and consider the sequence p = (p1, p2,...) of first digits of the orbit of θ: { x, θ(x), θ2(x), ...}. That is, pk = ⎣θk(x)/br⎦∈D, where r=⎣log10θk(x)⎦. The purpose of this paper is to investigate the dynamical properties of the sequence p.

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Reference Type: Journal Article

Subject Area(s): Dynamical Systems